Olympiad problems

2009 Today's Calculation Of Integral, 448

Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$

2010 Today's Calculation Of Integral, 619

Tags: calculus , integration , function , trigonometry , calculus computations , logarithm

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

2005 Today's Calculation Of Integral, 32

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

2010 Today's Calculation Of Integral, 652

Tags: calculus , integration , calculus computations

Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

2020 Jozsef Wildt International Math Competition, W33

Tags: calculus , integration , function

Let $p\in\mathbb N,f:[0,1]\to(0,\infty)$ be a continuous function and $$a_n=\int^1_0x^p\sqrt[n]{f(x)}dx,n\in\mathbb N,n\ge2.$$ Demonstrate that: a) $\lim_{n\to\infty}a_n=\frac1{p+1}$ b) $\lim_{n\to\infty}((p+1)a_n)^n=\exp\left((p+1)\int^1_0x^p\ln f(x)dx\right)$ [i]Proposed by Nicolae Papacu[/i]

2007 AIME Problems, 7

Tags: function , ceiling function , floor function , calculus , integration , logarithm

Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)

2008 Harvard-MIT Mathematics Tournament, 4

Tags: quadratic , limit , calculus , calculus computations , logarithm

([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.

2005 ISI B.Math Entrance Exam, 6

Tags: algebra , polynomial , integration , calculus , geometry , calculus computations

Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

2009 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , limit , trigonometry , derivative , function , real analysis

Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]

2024 CMIMC Integration Bee, 14

Tags: calculus , integration

\[\int_1^\infty \frac{\lfloor x \rfloor}{9x^3-x}\mathrm dx\] [i]Proposed by Robert Trosten and Connor Gordon[/i]

2015 District Olympiad, 2

Tags: integral , real analysis , calculus , integration

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

2011 Today's Calculation Of Integral, 755

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

2011 ISI B.Stat Entrance Exam, 1

Tags: calculus , inequalities , rearrangement inequality , n-variable inequality

Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that \[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]

2009 Today's Calculation Of Integral, 397

Tags: calculus , integration , parabola , quadratic , function , analytic geometry , conic

In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis

2007 Today's Calculation Of Integral, 207

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]

2007 Today's Calculation Of Integral, 204

Tags: calculus , integration , trigonometry , special factorizations , calculus computations

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

Today's calculation of integrals, 882

Tags: calculus , integration , limit , calculus computations , riemann sum , logarithm

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2007 Today's Calculation Of Integral, 243

Tags: calculus , integration , function , calculus computations , geometry

A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.

2009 VTRMC, Problem 7

Tags: calculus , differential equation

Does there exist a twice differentiable function $f:\mathbb R\to\mathbb R$ such that $f'(x)=f(x+1)-f(x)$ for all $x$ and $f''(0)\ne0$? Justify your answer.

2021 CMIMC Integration Bee, 11

Tags: calculus , integration

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

1969 German National Olympiad, 6

Tags: polynomial , analysis , calculus , algebra

Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.

2011 Today's Calculation Of Integral, 726

Tags: calculus , integration , calculus computations , logarithm , geometry

Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

1994 Irish Math Olympiad, 1

Tags: ireland , quadratic , calculus , integration , algebra , number theory

Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$

2014 IPhOO, 4

Tags: calculus , integration , probability , expected value

A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $. If the number of photographs taken is huge, find $ k $. That is: what is the time-average of the distance traveled divided by $ h $, dividing by $h$? $ \textbf {(A) } \dfrac{1}{4} \qquad \textbf {(B) } \dfrac{1}{3} \qquad \textbf {(C) } \dfrac{1}{\sqrt{2}} \qquad \textbf {(D) } \dfrac{1}{2} \qquad \textbf {(E) } \dfrac{1}{\sqrt{3}} $ [i]Problem proposed by Ahaan Rungta[/i]

2005 VJIMC, Problem 3

Tags: limit , calculus , integration

Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit $$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$